If the phrase is 'Don't you ever let out', this can be used to ask someone, 'Don't you ever relax and have fun?'. This was used on campus when I was young (i.e. many years ago), so I would not be surprised if it has taken on a new meaning now.
If the correct phrase is 'Don't you ever let up?' then it would make a world of difference. 'Let up' is very useful phrase that can be used for a wide variety of reasons.
'Don't you ever let up?' means 'Don't you ever stop?', or 'Don't you ever give people a break?' You would normally say this to someone who is annoying you because they are doing something repeatedly, e.g.:
always making sexist/racist/rude comments
always complaining about something
always pushing their view about politics/religion/their hobby
always behaving inappropriately
The term has wider application. For example, if it has been raining for days, people will start to ask, 'When is the rain going to let up?', meaning 'When is the rain going to stop.' you can make similar comments about
illness, e.g. a persistent cough
a long-term economic downturn
a boss who is always complaining about his workers
a politician who is always talking about the same thing